Solve the following linear programming problem.
\[
\begin{array}{ll}
\text { Maximize: } & z=x+6 y \\
\text { subject to: } & 3 x+4 y \leq 12 \\
& 6 x+y \leq 12 \\
& x \geq 0, y \geq 0
\end{array}
\]
The maximum value is 18 .
The maximum occurs at the point
(Type an ordered pair. If the maximum occurs at more than one point, type either answer. Type an integer or a fraction.)

Step 1 ：The problem is a linear programming problem. The goal is to maximize the function \(z = x + 6y\) subject to the constraints \(3x + 4y ≤ 12\), \(6x + y ≤ 12\), and \(x, y ≥ 0\).

Step 2 ：To solve this problem, we can use the graphical method. This involves plotting the constraints on a graph and finding the feasible region. The maximum value of the function will occur at a vertex of the feasible region.

Step 3 ：However, since we want to maximize the function, we can multiply the coefficients of the objective function by -1 to convert the problem into a minimization problem.

Step 4 ：The Python code successfully solved the linear programming problem. The optimal solution is \(x = 0\), \(y = 3\), and the maximum value of the function is \(z = 18\). This matches the given maximum value in the question.

Step 5 ：Final Answer: The maximum value is \(\boxed{18}\) and it occurs at the point \(\boxed{(0, 3)}\).